Find the derivative of the following function using chain rule 1. $$ f(x)=(3 x+1)\left(4 x^{2}-2 x\right) $$

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To find the derivative of the function $$ f(x) = (3x + 1)(4x^2 - 2x) $$ using the **chain rule**, it's actually more appropriate to use the **product rule** since the function is a product of two functions. The product rule states that if you have two functions $$ u(x) $$ and $$ v(x) $$, then the derivative of their product is:

$$
(fg)' = f'g + fg'
$$

Let's apply this to the given function:

1. **Identify the functions:**
   $$
   u(x) = 3x + 1 \quad \text{and} \quad v(x) = 4x^2 - 2x
   $$

2. **Find the derivatives of $$ u(x) $$ and $$ v(x) $$:**
   $$
   u'(x) = \frac{d}{dx}(3x + 1) = 3
   $$
   $$
   v'(x) = \frac{d}{dx}(4x^2 - 2x) = 8x - 2
   $$

3. **Apply the product rule:**
   $$
   f'(x) = u'(x)v(x) + u(x)v'(x)
   $$
   $$
   f'(x) = 3(4x^2 - 2x) + (3x + 1)(8x - 2)
   $$

4. **Expand and simplify:**
   $$
   f'(x) = 12x^2 - 6x + (24x^2 - 6x + 8x - 2)
   $$
   $$
   f'(x) = 12x^2 - 6x + 24x^2 + 2x - 2
   $$
   $$
   f'(x) = (12x^2 + 24x^2) + (-6x + 2x) - 2
   $$
   $$
   f'(x) = 36x^2 - 4x - 2
   $$

**Final Answer:**
$$
f'(x) = 36x^2 - 4x - 2
$$
The derivative of $$ f(x) = (3x + 1)(4x^2 - 2x) $$ is $$ f'(x) = 36x^2 - 4x - 2 $$.

Find the derivative of the following function using chain rule 1. \( f(x)=(3 x+1)\left(4 x^{2}-2 x\right) \)

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